RE: [Swarm-Modelling] Re: Mixtures of Distributions, Vol 1 # 117

[Christopher J. Mackie]

Just a little addendum to Rick's message, since it mentions Excel.  If you're 
not aware, Excel 2003 has a bad bug in the RAND() function family that 
adversely affects anyone using random number generation. 
 
The bug affects only Office 2003 (they rewrote the random # generator for the 
version).
 
Details (such as they are) are available at 
 
http://support.microsoft.com/default.aspx?scid=kb;en-us;834691
 
although you'll get a more honest account by searching for 'Excel rand bug' in 
google.
 
Workarounds have been available for some time, but they're not entirely 
satisfactory: the resulting distributions don't include negative numbers, but 
do have some biases.  Fortunately, there's now a full patch available; it 
appears to solve the problem entirely.  You can get it via the OfficeUpdate 
web-based updater (officeupdate.microsoft.com) or directly at 
 
http://www.microsoft.com/office/ork/updates/2003/exc1101a.htm
 
If you're going to use Excel 2003 for anything that depends on random number 
generation, you need this patch in order to avoid some nasty surprises.
 
Hope this helps, --Chris

        -----Original Message----- 
        From: address@hidden on behalf of Rick Lightburn 
        Sent: Tue 4/13/2004 5:55 PM 
        To: address@hidden 
        Cc: 
        Subject: [Swarm-Modelling] Re: Mixtures of Distributions, Vol 1 # 117
        
        

        Paul Johnson wants to know if there is an "easy" (my quotes, not his) 
distribution from which the first parameter from among a mixture of betas has a 
distribution.  Probably not, but that shouldn't really matter. 
        
        Johnson is well to choose the beta for the proclivity of political 
orientation.   The resulting mixture distribution will have to be calculated by 
what used to be called 'brute-force' methods, and now might be called 
'computer-intensive' ones.  I don't think that this group of modelers would be 
dissuaded from computer-intensive methods, or find more value inherent in a 
closed-form for the mixture.
        
        I imagine that there is there some content-based motivation for 
assuming the distribution of the 'hyperparameter'.  (I'm not a political 
scientist, otherwise I could indicate something what it might be.)  I'd think 
the 'natural' thing to do would be that a priori the distribution of the 
parameter in, say, 2000 would be the observed distribution in 1996.
        
        Note that the mean of the 'state-level' beta is a/(a+b).  Therefore it 
might be useful to re-parameterize the 'state-level' distributions in terms of 
their means and something else (maybe b, but I haven't thought it through).  
Alternatively, it might make sense to fix a and b relatively, so that a/(a+b) = 
0.5, so that a=b, and then the family of distributions Johnson would be 
examining would be a one-parameter family.  If this common parameter were 
allowed to vary uniformly over the range (0.5, 1.5), say, then individual 
states would vary from highly polarized (when a=b~0.5) to reasonably cohesive 
and 'predictable' (when a=b~1.5).
        
        As for 'estimating' the hyperparameter, or the individual parameters, 
there is going to be a problem with identification:  I think there will be more 
parameters than observations, and only a very devout Bayesian would even 
contemplate such a problem.  Gregg Allenby, at Ohio State, is the key guru on 
the Bayesian analysis of mixture distributions, and if it ends up in a 
very-high dimensional integral, well, Allenby would be the resource on the 
MonteCarlo integrals that do such things. 
        
        But 'simulating' a mixtures of betas (even one with equal parameters), 
which is a very natural thing to do, shouldn't be all that difficult to code.  
(One could probably do it in Excel in under an hour.  Sam Savage, at Stanford, 
has an Excel add-in that would substantially facilitate that.)
        
        -----Original Message-----
        From: address@hidden
        Sent: Apr 13, 2004 2:00 PM
        To: address@hidden
        Subject: Modelling digest, Vol 1 #117 - 1 msg
        
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        Today's Topics:
        
           1. Can you help me with a probablility question about mixtures of 
distributions? (Paul Johnson)
        
        --__--__--
        
        Message: 1
        Date: Tue, 13 Apr 2004 11:30:27 -0500
        From: Paul Johnson <address@hidden>
        To: swarm-modelling <address@hidden>
        Subject: [Swarm-Modelling] Can you help me with a probablility question 
about mixtures of distributions?
        Reply-To: address@hidden
        
        Suppose there are 50 collections of agents. Think of these collections
        as districts in a political system.   For each collection, we have 1000
        agents.  I want each agent to have a meaningul parameter in the
        left-right political scale, and I'm thinking of using a Beta
        distribution because it is bounded and displays a wide variety of
        shapes.  I vary the parameters so that not all districts are exactly the
        same in political composition.   As a first take, I have the assumption
        that the distribution within each cluster has a "cluster-specific"
        parameter, a_k, representing the first beta parameter.  So the
        observations are B(a_k,b) for districts k=1...50.
        
        Question: is there a distribution from which to draw a_k so that the
        combined set of all agents has a known distribution?
        
        I've stumbled around a while and I find plenty of literature on Bayesian
        statistics and the Beta as a prior to the Binomail distribution, but I
        can't find anything about the more mundane simulation question "if I
        generate cases like so, what do I have?"
        
        What is the proper  literature to read?
        
        pj
        
        --
        Paul E. Johnson                       email: address@hidden
        Dept. of Political Science            http://lark.cc.ku.edu/~pauljohn
        1541 Lilac Lane, Rm 504                             
        University of Kansas                  Office: (785) 864-9086
        Lawrence, Kansas 66044-3177           FAX: (785) 864-5700
        
        
        
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